For \(t^{\prime }= 0\) , the \(W=4\)  leg cylinder shows significantly different tendencies. The most extensive studies [1]}, [2]}, [3]}, [4]} have been carried out primarily for the (assumed representative) value of \(x=1/8\) . Here the preferred CDW ordering vector is half what it was in the previous case, \(Q^{\vphantom{\dagger }}_{\rm CDW} = 2k^{\vphantom{\dagger }}_{\rm F} \equiv 2\pi x\) . Moreover, it seems that the SC correlations fall rapidly – most probably exponentially - with distance. (However, in this case, the SC correlation length, \(\xi ^{\vphantom{\dagger }}_{\rm SC}\)  is remarkably long - it is estimated in ref. [5]} that \(\xi ^{\vphantom{\dagger }}_{\rm SC} \approx 18\) ). There is again significant short-range SC and SDW order, with, as in Fig. REF a, the SC order being \(d\) -wave-like, and the SDW ordering vector satisfying the mutual comensurability condition, \(2Q^{\vphantom{\dagger }}_{\rm SDW} \equiv Q^{\vphantom{\dagger }}_{\rm CDW}\) . This, if extrapolated to the 2d limit, is suggestive [2]} of a commensurate unidirectional CDW insulating state, either with or without accompanying SDW order, with a significant but strictly short-range correlated tendency toward \(d\) -wave SC. Thinking of the peaks in the CDW as being a stripe of high doped hole density, this state, which arises naturally in Hartree-Fock calculations in 2d, is sometimes referred to as having “full stripes” – full, in the sense that since \(\lambda ^{\vphantom{\dagger }}_{\rm CDW} = 1/x\) , there is one doped hole per site along the length of each stripe. By contrast, the state with \(\lambda ^{\vphantom{\dagger }}_{\rm CDW} = {\textstyle {1 \over 2x}}\)  (\(Q^{\vphantom{\dagger }}_{\rm CDW} = 4\pi x\) ) is then referred to as ”half-filled” stripes.
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